One dimensional Willis-form equations can retain time synchronization under spatial transformations

This paper gives a theoretical proof that the one dimensional (1D) Willis-form equations with displacement coupling terms can retain time synchronization under spatial transformations. This is also supported by a numerical example that compares the distributions of the wave velocities of an inhomogeneous material calculated by the Willis-form equations and the classical elastodynamic equations, respectively. It further demonstrates that the classical elastodynamic equations are good approximations of the Willis-form equations only when the wave frequency is sufficiently high. These findings serve as additional evidence of the validity of the Willis-form equations for inhomogeneous media.